Less than 10 minutes after David Cameron ended his speech to MPs about how he proposed making sure the hacking scandal would never be repeated, the Murdoch jet took off. As it banked into the cloud, it might have been writing "crisis over" in the sky.

Last night, the Cameron team came back from Africa (where defeating criticism by ignoring it is a well-established political technique) with a strategy that for the first time in a fortnight seems to have got the prime minister onto the front foot. After more than two hours answering questions on the details of the phone-hacking scandal to a packed Commons, the bung on the lilo of outrage was successfully pulled with a debate on the important but intangible question of public confidence in the media. Members evaporated from the chamber as if they could already hear the final call for their holiday flights.

Cameron is not out of trouble yet. But with Rupert Murdoch's helpful testimony yesterday about his fondness for Gordon Brown, the prime minister successfully reminded the world that relations between politicians, the media and the police were not a problem that had only just materialised. He appealed for cross-party support to tackle the issue in the long term, replacing the bidding war between parties for support by a common front for restraint. That meant that if he pursued his attack on Cameron's judgement over Andy Coulson, Ed Miliband risked being left vulnerable to the charge of tribalism.

Cameron trailed some preoccupations that should worry the opposition. He announced a review of relations between the media and Whitehall officials that hints at early tensions between his government and civil servants. More alarmingly, he introduced an equivalence of media power between News International and the BBC that will no doubt have pleased Murdoch as he fastened his seatbelt for his return journey, but should alarm the rest of us. But he also proposed interesting ideas on regulation, such as treating assessment of concentration of media ownership as a continuing process rather than an event, and removing politicians from it altogether.

Cameron emerges looking stronger: backbenchers had been prepped with supportive points to make. But it was one better day after a lot of bad ones. As MPs depart for the summer, the ball is lying in the outfield. No doubt the prime minister is hoping the grass will grow tall and lush around it before the Commons comes back in September.

David Cameron rightly said that the need to root out the police corruption highlighted by the hacking affair demonstrates the need to take a broader look at the whole culture of policing in this country.

As he pointed out, the first step is rebuild the leadership of the Metropolitan police. The process of finding a new commissioner is to be accelerated and Cressida Dick took over today as John Yates's replacement in the vital counter-terrorism job. Dick was at the centre of the storm over the shooting of Jean Charles de Menezes but even the Justice4Jean campaign haven't said that she shouldn't get he job.

The argument that the next commissioner should be a woman is gathering ground in Whitehall and within the police as they learn more and more of the "cosy", hard-drinking, "wine and dine" culture that has flourished between the police and the tabloids. That may not work out, if only because all the likely candidates have some sort of baggage.

Cameron says that far more needs to be done as "there are too few – and arguably too similar – candidates for the top jobs". His solution is to open up the senior ranks of the police to people from other walks of life and to allow the recruitment of exceptional talent from overseas.

For decades the police force was built on the notion that the chief constable at the top had started life by walking the beat and had tasted life on the frontline. It is also a legal requirement that no British police force can be run by a foreign national.

What has Cameron got in mind? What he would really like to do is bring in a successful American police chief, in the mould of Bill Bratton who delivered in New York, to run a British force. Labour was not averse to this idea. David Blunkett brought over the former Boston police chief, Paul Evans, to run the Home Office's police improvement unit, but he could not be any more than a civil servant.

In the debate Cameron shackled this idea together with the introduction of elected police and crime commissioners from next May.

For the Conservatives, fresh talent and improved accountability will end police corruption. But Cameron needs to look at the history of the 1950s and 1960s when local councillors last had the power to hire and fire chief constables and there were policing scandals as a result in Nottingham and Brighton.

As Labour's Karen Buck pointed out in the Commons, what hope is there for the operational independence of the police when an elected mayor or police commssioner declares that a set of allegations are "codswallop", as Boris Johnson did in this case, before any serious investigation has got under way.

If we weren't clear before, we are now. Rather than looking forensically into News International's handling of the phone-hacking scandal, what David Cameron said in parliament today reveals his judicial inquiry will be looking to hold the press in general to account.

Let's leave to one side his stonewalling obfuscation over whether he discussed Rupert Murdoch's bid for BSkyB with executives of News International (although his refusal to deny it looks suspect given the full 26 recorded meetings), the rest of his performance in the debate that followed his statement gave cause for concern for other parts of the media ecosystem. Indeed, experts in the corporate speak beloved in Wapping would suggest that the key thing David Cameron is taking from the phone-hacking mess is that all journalists - even on the pesky "social media" no less - need reining in.

I need only refer you to a stunning passage that would otherwise be overlooked in justifiable outrage over his comments on BSkyB. Speaking more in sorrow than in anger, the prime minister said: "We put on the back burner for too long the issue of how to regulate the media."

Relations between a cowed political class and the media should be looked at more generally, he went on: "And not just at NI but also ... the BBC, the Independent and the Guardian. This is a cathartic moment to sort it out and put it on a proper footing."

So, the flood used to clean the Augean stables that have almost become a cliche to describe the mess at News International will clear out the rest of journalism. Now that would be a political triumph for David Cameron.

The three former senior journalists to sit on Lord Justice Leveson's inquiry - Elinor Goodman, George Jones and Sir David Bell - hail from C4, the Telegraph and the FT respectively. Cameron described them as having "complete independent from interested parties". Let's hope their hearings - to be held in public, thank goodness - will show similar independence from such political gerrymandering.

• This live panel on today's hacking debate in the Commons will be updated throughout the day with entries from Alan Travis on the police, Jane Martinson on the media and Anne Perkins on the politics.